Optimality of Curtiss Bound on Poincare Multiplier for Positive Univariate Polynomials

Let $f$ be a monic univariate polynomial with non-zero constant term. We say that $f$ is positive if $f(x)$ is positive over all $x\ge 0$. If all the coefficients of $f$ are non-negative, then $f$ is trivially positive. In 1883, Poincaré proved that$f$ is positive if and only if there exists a monic polynomial $g$ such that all the coefficients of $gf$ are non-negative. Such polynomial $g$ is called a Poincaré multiplier for the positive polynomial $f$. Of course one hopes to find a multiplier with smallest degree. This naturally raised a challenge: find an upper bound on the smallest degree of multipliers. In 1918, Curtiss provided such a bound. Curtiss also showed that the bound is optimal (smallest) when degree of $f$ is 1 or 2. It is easy to show that the bound is not optimal when degree of $f$ is higher. The Curtiss bound is a simple expression that depends only on the angle (argument) of non-real roots of $f$. In this paper, we show that the Curtiss bound is optimal among all the bounds that depends only on the angles.